On orders of two transformation semigroups of the boolean

I.V. Livinsky; T.G. Zhukovska

doi:10.15330/cmp.6.2.317-319

I.V. Livinsky Taras Shevchenko National University, 64/13 Volodymyrska str., 01601, Kyiv, Ukraine
T.G. Zhukovska Lesya Ukrainka East European National University, 13 Voli avenue, 43025, Lutsk, Ukraine

https://doi.org/10.15330/cmp.6.2.317-319

Keywords: semigroup, order-preserving transformation, order-decreasing transformation, monotone boolean functions

Published online: 2014-12-27

Abstract

We consider the semigroup $\mathcal{O}(\mathcal{B}_n)$ of all order-preserving transformations $\varphi : \mathcal{B}_n \rightarrow \mathcal{B}_n$ of ordered by inclusion boolean $\mathcal{B}_n$ of $n$-element set (i.e. such transformations that $A \subseteq B$ implies $\varphi(A) \subseteq \varphi(B)$) and its subsemigroup $\mathcal{C}(\mathcal{B}_n)$ of those transformations for which $\varphi(A) \subseteq A$ for all $A \in \mathcal{B}_n$. Orders of these semigroups are calculated.